Transcript Lecture 2 * Energy Functions - LCQB

Structural
Bioinformatics
Elodie Laine
Master BIM-BMC Semester 3, 2016-2017
Laboratoire de Biologie Computationnelle et Quantitative (LCQB)
e-documents: http://www.lgm.upmc.fr/laine/STRUCT
e-mail: [email protected]
Lecture 2 – Energy
Functions
Elodie Laine – 11.10.2016
Context
How can we determine a protein 3-dimensional coordinates ?
Experimental techniques
(X-ray crystallography, NMR,
cryo-EM…)
In silico prediction
(bioinformatics methods)

Huge number of possible
conformations
 The native structure generally
corresponds to one of these
conformations: the free energy
minimum
 Time necessary for a protein to
adopt its native structure:
between 1 ms and 1 s
Elodie Laine – 11.10.2016
The thermodynamic hypothesis (Anfinsen's Dogma)
The native structure of a protein
corresponds to minimum of free energy
The free energy difference between the
native state and the ensemble of
denatured conformations is 20 à 60 kJ/mol
(5-15 kcal/mol).
Elodie Laine – 11.10.2016
The thermodynamic hypothesis (Anfinsen's Dogma)
 Uniqueness: the sequence does not have any other configuration with a
comparable free energy.
 Stability: small changes in the surrounding environment cannot give rise to
changes in the minimum configuration.
 Kinetical accessibility: the folding of the chain must not involve highly
complex changes in the shape (like knots or other high order conformations).
Is this assumption still admitted?
What type of proteins challenge it?
Elodie Laine – 11.10.2016
Free energy
The protein in solution is viewed as a statistical ensemble.
G  H  TS
Gibbs or Helmholtz
free energy
Energy available
for
thermodynamic
work
Enthalpy or internal
energy
Internal
interactions
Entropy by
temperature
Hydrophobic
effect and
conformational
entropy
Elodie Laine – 11.10.2016
Free energy
-
0
+
-TS
Conformational
entropy
Favorable free
energy of
folding is a net
result of
thermodynamic
forces
H
Internal
interactions
-TS
Hydrophobic
effect
G
Folding
Elodie Laine – 11.10.2016
Interatomic interactions
Amino acids of a protein are joined by covalent bonding interactions (primary structure).
The 3D fold is stabilized by non-bonding interactions (tertiary structure):
• Electrostatic Interactions (5 kcal/mol)
• Hydrogen-bond Interactions (3-7 kcal/mol)
• Van Der Waals Interactions (1 kcal/mol)
• Hydrophobic Interactions (< 10 kcal/mol)
Interactions are treated
differently depending on the
number of amino acids along the
sequence separating the
interacting pair.
Elodie Laine – 11.10.2016
Energy functions
Semi-empirical potentials

analytical forms describing
interactions which parameters
are fitted to:
• experimental data
• quantum mechanics
calculations
Statistical potentials

analytical forms describing
interactions which parameters
are derived from a database of
known structures

sequence-structure association
frequencies converted to free
energies
Elodie Laine – 11.10.2016
Semi-empirical potentials
These potentials are analytical expressions that describe inter-atomic interactions.
They represent molecular mechanics models of proteins containing:
- some chosen interactions
- a chosen functional that describes and links them
Schrödinger
Newton
Born-Oppenheimer
Additivity
Transferability
Empirical
Their general form is:
E = Ebond + Eangle + Etorsion + Enon-bonded + Eothers
Elodie Laine – 11.10.2016
Molecular mechanics energy

An example: AMBER force field
O(N)
O(N2)
O(NlogN)
Many more… CHarMM, OPLS…
Elodie Laine – 11.10.2016
Bonded interactions
How can we represent the variation of the energy corresponding to a
covalent bond stretching and bending?
Covalent bond
potential energy:
which function to model
this curve ?
Elodie Laine – 11.10.2016
Bonded interactions

Morse potential
E  De (1  e  a ( r  re ) ) 2
r: interatomic distance
re: equilibrium bond distance
De: well-depth
a: controls the width of the potential

Harmonic potential
E
1
k (r  re ) 2
2
r: interatomic distance
re: equilibrium bond distance
k: spring force constant
Elodie Laine – 11.10.2016
Bonded interactions
Bond
Angle
Dihedral
Elodie Laine – 11.10.2016
Bonded interactions
Bond
Angle
b0
Dihedral
Elodie Laine – 11.10.2016
Bonded interactions
Bond
Angle
θ0
Dihedral
Elodie Laine – 11.10.2016
Bonded interactions
Bond
Angle
Φ0
Dihedral
Elodie Laine – 11.10.2016
Non-bonded interactions
Electrostatics
van der Waals
+
-
Coulomb interaction between
single point charges
Elodie Laine – 11.10.2016
Non-bonded interactions
Electrostatics
van der Waals
+
+
Coulomb interaction between
single point charges
Elodie Laine – 11.10.2016
Non-bonded interactions
Electrostatics
van der Waals
Hard core repulsion between
close atoms
Elodie Laine – 11.10.2016
Non-bonded interactions
Electrostatics
van der Waals
Weak dipole attraction between
distant atoms
Elodie Laine – 11.10.2016
Lennard-Jones potential
van der Waals
Elodie Laine – 11.10.2016
Solvent models
Explicit
Implicit
Elodie Laine – 11.10.2016
Parametrization
Determining parameter values that best fit the force field and lead to
the most accurate energy estimates is not trivial.
Accuracy
?
Simplicity
Generality
Parameters are fitted to experimental data (spectroscopy, small molecular
crystals…) or quantum mechanics calculations. They are computed for a certain
type of molecules (proteins, nucleic acids…) and may not be transferable.
Elodie Laine – 11.10.2016
Pair vs multi-body potentials
Coulombic and van der Waals potentials are summed over pairs of atoms.
How do we account for the influence of all the other particles in the system ?
Pairs: N(N-1)/2
Triplets: N(N-1)(N-2)/6
Effective potentials: account for the presence of the other entities through
parametrization. An effective pair potential does not reflect the « true »
interaction energy between two isolated atoms but is parametrized so as to
include the effect of the other atoms in the energy of the pair.
Elodie Laine – 11.10.2016
Energy functions
Semi-empirical potentials

analytical forms describing
interactions which parameters
are fitted to:
• experimental data
• quantum mechanics
calculations
Statistical potentials

analytical form describing
interactions which parameters
are derived from a database of
known structures

sequence-structure association
frequencies converted to free
energies
Elodie Laine – 11.10.2016
Statistical mechanics
Proteins adopt an ensemble of conformations in solution. Not every protein
in a large group of them has the lowest energy. The energies are random but
they obey certain statistical laws based on the Boltzmann distribution.
The probability of observing a given
conformation Ci is:
Boltzmann coefficient
P(Ci ) 
exp(  E (Ci ) / kT )
 exp(  E(Ci ) / kT )
partition function
Elodie Laine – 11.10.2016
Statistical potentials: worflow
1/ Data
Collection
• Low sequence similarity (<25 %) to avoid bias
• High resolution (<2 Å) and high quality structures
• Minimum critical size of the database to derive significant statistics
2/ Sequences and structures
subdivision
1/ Parametrization of the
potential
Distance , torsion, hydrophobicity…
Elodie Laine – 11.10.2016
Distance potentials
Potential of mean force w(2) between two particles at positions r1 and r2:
 
exp[  w (r1 , r2 ) / kT ] 
( 2)
 
P ( 2) (r1 , r2 )


P (1) (r1 ) P (1) (r2 )
P(1)(r1): probability of one particle being in position r1
P(2)(r1,,r2): probability of the two particles being in respective positions r1 and r2
Potential of mean force W(2) between two particles of types s1 and s2 at
positions r1 and r2:
 
exp[ W ( 2) (r1 , r2 ; s1 , s2 ) / kT ] 
 
P ( 2) (r1 , r2 | s1 , s2 )


P (1) (r1 | s1 ) P (1) (r2 | s 2 )
Elodie Laine – 11.10.2016
Distance potentials
Potential of mean force W(2) of a system with different types of particles
compared to a reference system with only one type of particles:
 
 
 
W ( 2) (r1 , r2 ; s1 , s2 )  W ( 2) (r1 , r2 ; s1 , s2 )  w( 2) (r1 , r2 )
Free
energy
denatured
state
Estimation of the potential of mean force W(n) for the entire system by
summing over all pairwise interactions:
W
( n)


(r1 ,..., rn ; s1 ,..., sn ) 
( 2)  
W
 (ri , rj ; si , s j )
n
i , j 1;i  j
s1,…,sn: amino acid types
r1,,…,rn: distance between amino acid residues
Elodie Laine – 11.10.2016
Distance potentials
Potential of mean force W(2) of a system with different types of particles
compared to a reference system with only one type of particles:
In practice, how do
 get the probabilities
  the free
( 2 ) we
( 2 )   P and from
( 2 ) there

W
(
r
,
r
;
s
,
s
)

W
(
r
,
r
;
s
,
s
)

w
(
r
1 2 1 2
1 2 1 2
1 , r2 )
energy?

Free
denatured

F (r12 | s1 , s2 )
( 2)
energy
W (r ; s , s )  kT ln
state

12
1
2
F (r12 )
Distances are computed between Cα, Cβ or side-chain centroïd Cμ or all atoms.
Estimation of the potential of mean force W(n) for the entire system by
It is possible to introduced different levels of refinement by :
summing over all pairwise interactions:
๏ combining frequencies of neighboring regions (potential smoothing)
n
๏ computing frequencies
separately
for aas close
in the sequence (2-8 aas) and

( n) 
( 2)  
W8 aas
(r1 ,...,
rn ; s1 ,..., sn ) potentials)
W (ri , rj ; si , s j )
aas further than
(local/non-local

i , j 1;i  j
s1,…,sn: amino acid types
r1,,…,rn: distance between amino acid residues
Elodie Laine – 11.10.2016
Energy functions
Semi-empirical potentials
-Physical
interpretation of
the force field
terms/parameters
-High cost to
accurately
account for
solvent & entropic
effects
Statistical potentials
-Can be adapted
to a coarsegrained
representation of
the protein
-Implicitly include
solvent & entropic
effects
-No obvious
physical
interpretation
-Dependence on
some
characteristics of
the database
Elodie Laine – 11.10.2016
Energy functions evaluation
The performances of an energy function can be evaluated using decoy sets,
generated by:
 Simulations of protein folding
 Comparative modeling
 Sequence inversion
Decoy sets must be large, contain realistic and representative structures.
A good energy function must:
 Assign the lowest energy to the native structure
 Discriminate the native structure from the decoys
 Display decrease of the energy of non-native structures as they become
more and more similar to the native structures
Low RMSD, high coverage of native contacs
Elodie Laine – 11.10.2016
Conclusion
• The native state of a protein corresponds to the global free
energy minimum
• The stability of a protein structure can be expressed as a
molecular mechanics potential energy or a potential of mean
force (free energy)
• The energy of stability can be evaluated as a sum over
physiscal terms describing the interatomic forces or over
statistical terms describing the frequencies of co-occurrences
of residue/atom conformations
• Parameters are either fitted to experimental data or more
sophiscticate calculations, or derived from databases
• Energy functions must be carefully evaluated
Elodie Laine – 11.10.2016